R2SM: a package for the analytic computation of the R2 Rational terms in the Standard Model of the Electroweak interactions

TL;DR

The paper presents an analytic FORM-based package for computing the $R_2$ rational terms in 1-loop electroweak SM amplitudes, adopting a $d=4+\epsilon$ framework with the OPP decomposition to separate $R_2$ from the Cut-Constructible part. It constructs $R_2$ effective vertices from all up-to-4-point generic diagrams using generic SM fields, then dresses them with physical particles via automated do-loop routines, while supporting $R_\xi$ gauges and multiple dimensional regularization schemes. A key validation is the gauge-invariance check of the total $R$ contribution to the $H \to \gamma\gamma$ amplitude at one loop, reinforcing the correctness of the approach. The work provides a modular, reusable tool and a foundational step toward fully automated analytic calculation of $R_2$ vertices for arbitrary Lagrangians in gauge theories.

Abstract

The analytical package written in FORM presented in this paper allows the computation of the complete set of Feynman Rules producing the Rational terms of kind R2 contributing to the virtual part of NLO amplitudes in the Standard Model of the Electroweak interactions. Building block topologies filled by means of generic scalars, vectors and fermions, allowing to build these Feynman Rules in terms of specific elementary particles, are explicitly given in the Rxi gauge class, together with the automatic dressing procedure to obtain the Feynman Rules from them. The results in more specific gauges, like the 't Hooft Feynman one, follow as particular cases, in both the HV and the FDH dimensional regularization schemes. As a check on our formulas, the gauge independence of the total Rational contribution (R1 + R2) to renormalized S-matrix elements is verified by considering the specific example of the H --> gamma-gamma decay process at 1-loop. This package can be of interest for people aiming at a better understanding of the nature of the Rational terms. It is organized in a modular way, allowing a further use of some its files even in different contexts. Furthermore, it can be considered as a first seed in the effort towards a complete automation of the process of the analytical calculation of the R2 effective vertices, given the Lagrangian of a generic gauge theory of particle interactions.

Figures (5)

• Figure 1: Non null contributions to the ss, vs, vv and ff${\mathrm R_2}$ effective vertices in the generalized $R_\xi$ gauges, with generic finite $\xi$, $\xi_Z$, $\xi_A$. The corresponding analytical formulas associated to selected topologies of each generic diagram are included in the files $\texttt{xxxxgentop.h}$, with $\texttt{xxxx}$ = $\texttt{ss}$, $\texttt{vs}$, $\texttt{vv}$ and $\texttt{ff}$, respectively.
• Figure 2: Non null contributions to the sff, vff, sss, vss, svv and vvv${\mathrm R_2}$ effective vertices in the generalized $R_\xi$ gauges, with generic finite $\xi$, $\xi_Z$, $\xi_A$. The corresponding analytical formulas associated to selected topologies of each generic diagram are included in the files $\texttt{xxxxgentop.h}$, with $\texttt{xxxx}$ = $\texttt{sff}$, $\texttt{vff}$, $\texttt{sss}$, $\texttt{vss}$, $\texttt{svv}$ and $\texttt{vvv}$, respectively. For all diagrams including a $\texttt{vvvv}$ vertex see the comments in Fig. \ref{['figura2']}.
• Figure 3: Non null contributions to the ssss, ssvv and vvvv${\mathrm R_2}$ effective vertices in the generalized $R_\xi$ gauges, with generic finite $\xi$, $\xi_Z$, $\xi_A$. The corresponding analytical formulas associated to selected topologies of each generic diagram are included in the files $\texttt{xxxxgentop.h}$, with $\texttt{xxxx}$ = $\texttt{ssss}$, $\texttt{ssvv}$ and $\texttt{vvvv}$, respectively. For all diagrams including at least a $\texttt{vvvv}$ vertex see the comments in Fig. \ref{['figura2']}.
• Figure 4: 2-,3- and 4-point 1-particle irreducible topologies explicitly considered for the calculation of the $\rm R_2$ contributions included in the xxxxgentop.h files. The symbols $e_j$ (ext$j$fla in our files), with $j$ = 1,2,3,4, denote external particles. Two indices are associated to each internal particle $i_k$ (k = 1,2,3,4), corresponding to the two vertices to which it is connected. So $e_j i_k$ (e$j$int$k$fla in our files) denotes the $k$-th internal particle incoming in the vertex in which also the $j$-th external particle is entering. According to the notation used to write down the Feynman rules, all momenta in all vertices are supposed to be incoming. Different topologies have been obtained from these ones by non-cyclic permutations of the external particles, together with their momenta and their Lorentz indices (omitted for simplicity in this figure), if present, for each fixed configuration of the internal ones.
• Figure 5: Diagrams in terms of generic fields, including a vvvv vertex, considered separately in our work. The charged vector bosons are denoted by $W^\pm$ whereas the neutral vector bosons are denoted by $V$. The first two diagrams are grouped together since they give rise to the same $R_2$ formulas in the ${\texttt{xxvvgentop.h}}$ files, as well as the second two.